Naty's answer is close but since you're looking for the "correct distortion of a cylinder", you should use an actual cylinder. Illustrator has very limited (and sometimes buggy) 3D tools available, but this is a case where it has just what you need.

In short, you should 3D extrude a circle and map the text around the cylinder that is created. To do this:

1. Create your text, make it a Symbol

To do that, just select the text and either drag it to the Symbols pane or click the New Symbol button.

2. Create a circle, 3D Extrude it

Select the circle, use Effect > 3D > Extrude & Bevel...

3. Map your symbol to the face of the cylinder

Click the Map Art... button, select the correct surface (the rectangular one), and apply your symbol

4. Set the "Surface" to "No Shading"

This is a dropdown in the "3D Extrude & Bevel Options"

And you're done!